A useful line of analysis is to consider the effect of scale changes for creatures
which are
similar in shape and only differ in scale. As the scale of an animal increases the body
weight and
volume increase with the cube
of scale. The volume of blood flow required to feed that bulk also increases with the cube
of scale. The cross sectional area of the arteries and the veins required to carry that
blood flow only increases with the square of scale. There are other area-volume relationships
which impose limitations on creatures. Some of those area-volume constraints, including the above one, are:

The surface area of the lungs for absorbing oxygen and evacuating carbon dioxide

The surface area of the intestines for absorbing food nutrients

The surface area of the skin for dispersing heat generated within the body

The cross sectional area of the arteries, capillaries and the veins for carrying
the blood required to transmit the oxygen, carbon dioxide, food nutrients and excess heat.

Thus to compensate for the body needs which increase with the cube of scale but
the areas increase with only the
square of scale the average blood flow velocity must
increase linearly with scale. Blood flow velocity is driven by pressure differences.
The pressure
difference must be great enough to carrying the blood flow to the top of the creature and great enough
to overcome the resistance in the arteries and veins to the flow. The pressure required to
pump blood from the heart to the top of the creature is proportional to scale.
The pressure difference required to overcome the resistance to flow through the arteries into
the capillaries and back again through the veins is more difficult to characterize in terms of scale.
The greater cross sectional area reduces the resistance but the long length increases resistance. The
net result of these two scale influences seems to be that the pressure difference required to
drive the blood through the bulk of the creature is inversely proportional to scale. The
pressure difference imposed would be the maximum of the two required pressure differences.

Shown below are the typical blood pressures for creatures of different scales.

Blood Pressure versus Height and Weight for Various Creatures

Creature

BloodPressure(mm Hg)

Height ofHead AboveHeart(mm)

Weight (grams)

Human

120

500

90000

Cow

157

500

800000

Duck

162

100

2000

Cat

129

100

2000

Guinea Pig

60

25

100

Goat

98

400

30000

Pig

128

200

150000

Monkey

140

200

5000

Dog

120

200

5000

Turkey

193

300

15000

Frog

24

25

50

Giraffe

300

3000

900000

Snake

55

25

100

The linear regression of the logarithm of pressure on the logarithm of height
yields the following result:

*log(Pressure) = 1.203 + 0.377*log(Height)
R2 = 0.675

The linear regression of the logarithm of pressure on the
logarithm of weight yields:

*log(Pressure) = 1.45 + 0.154*log(Weight)
R2 = 0.619

If blood pressure were proportional to scale then the coefficient for *log(Height) would be
1.0 and for *log(Weight) would be 0.333 since weight to proportional to the cube of scale.
The regression coefficients are not close to the theoretical values but they are of the
proper order of magnitude for accepting blood pressure as being
proportional to scale.

The volume of the heart of a creature is proportional to the cube of scale.
The volume
of the blood to be moved is also proportional to the cube of scale. From the previous
analysis the flow velocity is proportional to scale. Therefore the time required to evacuate
the heart's volume is proportional to scale. This means that the heartbeat rate is inversely
proportional to scale. The following table gives the heart rates for a number of creatures.

Heartbeat Rates of Animals

Creature

Average Heart Rate(beats per minute)

Weight(grams)

Human

60

90000

Cat

150

2000

Small dog

100

2000

Medium dog

90

5000

Large dogs:

75

8000

Hamster

450

60

Chick

400

50

Chicken

275

1500

Monkey

192

5000

Horse

44

1200000

Cow

65

800000

Pig

70

150000

Rabbit

205

1000

elephant

30

5000000

giraffe

65

900000

large whales

20

120000000

A regression of the logarithm of heart rate on the logarithm of weight yields the following
equation:

*log(heart rate) = 2.89 - 0.202*log(Weight)

If heart rate were exactly inversely proportion to scale the coefficient for *log(weight)
would be -0.333. This is because scale is proportional to the cube root of weight.
The coefficient of -0.2 indicates that the heart rate is given an equation of the form

heart rate = A(Scale)bwhere b= -0.6
&nbsp:

One salient hypothesis is that the animal heart is good for a fixed number of beats. This
hypothesis can be tested by comparing the product of average heart rate and longevity for
different animals. Because the heart rate is in beats per minute and longevity is in years the
number of heart beats per lifetime is about 526 thousand times the value of the product. The
data for a selection of animals are:

Lifetime Heartbeats and Animal Size

Weight

Heart Rate

Longevity

Product

Lifetime Heartbeats

Creature

(grams)

(/minute)

(years)

(billions)

Human

90000

60

70

4200

2.21

Cat

2000

150

15

2250

1.18

Small dog

2000

100

10

1000

0.53

Medium dog

5000

90

15

1350

0.71

Large dogs

8000

75

17

1275

0.67

Hamster

60

450

3

1350

0.71

Chicken

1500

275

15

4125

2.17

Monkey

5000

190

15

2850

1.50

Horse

1200000

44

40

1760

0.93

Cow

800000

65

22

1430

0.75

Pig

150000

70

25

1750

0.92

Rabbit

1000

205

9

1845

0.97

elephant

5000000

30

70

2100

1.1

giraffe

900000

65

20

1300

0.68

large whale

120000000

20

80

1600

0.84

Although the lack of dependence is clear visually the confirmation in terms of
regression analysis is:

*log(Lifetime Heartbeats) = 9.006 - 0.0046*log(Wt)
R² = 0.0018

The t-ratio for the slope coefficient is an insignificant 0.15, confirming that there is
no dependence of lifetime heartbeats on the scale of animal size.

If a heart is good for just a fixed number of beats, say one billion, then heart longevity is this
fixed quota of beats divided by the heart rate. From the above equation for heart rate,
lifespan (limited by heart function) would be proportional to scale raised to the 0.6 power.

Lifespan = B*(Scale)0.6which in term of animal weight would be
Lifespan = C*(Weight)0.2

The data for testing this deduction are:

Lifespan versus Weight for Various Creatures

Creature

Weight (grams)

Life Span(years)

Human

90000

70

Cow

800000

22

Duck

2000

10

Cat

2000

15

Guinea Pig

100

5

Goat

30000

15

Pig

150000

25

Monkey

5000

25

Dog

5000

15

Turkey

15000

5

Frog

50

3

Giraffe

900000

25

Snake

100

10

For the data in the above table, admittedly very rough and sparse, the regression of the
logarithm of the lifespan on the
logarithm of weight gives

*log(Lifespan) = 0.970 + 0.191*log(Weight)
R² = 0.527

Thus the net effect of scale on animal longevity is positive. Taking into account that weight is
proportional to the cube of the linear scale of an animal the above equation in terms of
scale would be

*log(Lifespan) = 0.970 + 0.573*log(Scale)

This says that if an animal is built on a 10 percent larger scale it will have a 6 percent
longer lifespan.